*3.1. New Similarity Transformation*

A new dimensionless time variable *τ* need to be introduced as follows (Zainal et al. [59])

$$
\pi = \frac{a}{1 - bt} t \tag{18}
$$

while the similarity variables (10) are replaced by

$$\eta = \left(\frac{a}{\nu\_f(1-bt)}\right)^{1/2} y\_r \,\psi = \left(\frac{a\nu\_f}{1-bt}\right)^{1/2} \ge f(\eta, \tau),\\N = \left(\frac{a}{\nu\_f(1-bt)}\right)^{1/2} \frac{a}{(1-bt)} \ge h(\eta, \tau),\\\theta(\eta, \tau) = \frac{T-T\_{\infty}}{T\_w - T\_{\infty}} \tag{19}$$

By applying Equations (18) and (19) in Equations (1)–(3) and (9), the new transformed differential equations are attained

$$\frac{\mu\_{\rm mf}/\mu\_{f}}{\rho\_{\rm mf}/\rho\_{f}}(1+K)\frac{\partial^{3}f}{\partial\eta^{3}}+f\frac{\partial^{2}f}{\partial\eta^{2}}-\left(\frac{\partial f}{\partial\eta}\right)^{2}+1-A\left(\frac{\partial f}{\partial\eta}+\frac{1}{2}\eta\frac{\partial^{2}f}{\partial\eta^{2}}-1\right)+\frac{K}{\rho\_{\rm mf}/\rho\_{f}}\frac{\partial h}{\partial\eta}-(A\tau+1)\frac{\partial^{2}f}{\partial\eta\partial\tau}=0\tag{20}$$

$$\frac{1}{\rho\_{\rm mf}/\rho\_f} \left( \frac{\mu\_{\rm mf}}{\mu\_f} + \frac{\mathcal{K}}{2} \right) \frac{\partial^2 h}{\partial \eta^2} + f \frac{\partial h}{\partial \eta} - \frac{\partial f}{\partial \eta} h - \frac{A}{2} \left( 3h + \eta \frac{\partial h}{\partial \eta} \right) - \frac{\mathcal{K}}{\rho\_{\rm mf}/\rho\_f} \left( 2h + \frac{\partial^2 f}{\partial \eta^2} \right) - (A\tau + 1) \frac{\partial h}{\partial \tau} = 0 \tag{21}$$

$$\frac{1}{\left(\Pr\left(\rho\mathbb{C}\_{p}\right)\_{\text{hfv}}/\left(\rho\mathbb{C}\_{p}\right)\_{f}\right)}\left(\frac{k\_{\text{hfv}}}{k\_{f}}+\frac{4}{3}\text{R}d\right)\frac{\partial^{2}\theta}{\partial\eta^{2}}+f\frac{\partial\theta}{\partial\eta}-2\frac{\partial f}{\partial\eta}\theta-\frac{A}{2}\left(3\theta+\eta\frac{\partial\theta}{\partial\eta}\right)-\left(A\tau+1\right)\frac{\partial\theta}{\partial\tau}=0\tag{22}$$

and the conditions become

$$\begin{aligned} f(0,\tau) &= 0, \,\frac{\partial f}{\partial \eta}(0,\tau) = \lambda, \, h(0,\tau) = -n \frac{\partial^2 f}{\partial \eta^2}(0,\tau), \,\theta(0,\tau) = 1, \\\ &\frac{\partial f}{\partial \eta}(\eta,\tau) \to 1, \, h(\eta,\tau) \to 0, \, \theta(\eta,\tau) \to 0 \text{ as } \eta \to \infty \end{aligned} \tag{23}$$
